Complex analysis method of steepest descent

How does the steepest descent method work?

Method of Steepest Descent
The main idea of the descent method is that we start with a starting point of x, try to find the next point that's closer to the solution, iterate over the process until we find the final solution.
For example, at step k, we are at the point ��(��)..

What is steepest descent method in chemistry?

3.3.
The steepest descent approach is to find smaller values of f by successively following directions in which f decreases.
As we have seen in the proof of the first-order necessary condition, − ∇ f provides such a direction..

What is the method of steepest descent formula?

Then the steepest descent directions from xk and xk+1 are orthogonal; that is, ∇f(xk) \xb7 ∇f(xk+1)=0. ) = −∇f(xk+1) \xb7 f(xk)=0.
That is, the Method of Steepest Descent pursues completely independent search directions from one iteration to the next..

What is the method of steepest descent in complex analysis?

In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase..

What is the method of steepest descent in physics?

The method of steepest descent, also known as the saddle-point method, is a natural development of Laplace's method applied to the asymptotic estimate of integrals of analytic functions..

What is the steepest descent algorithm used for?

In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function..

What is the steepest descent method in data mining?

In some literature, such as this and this, steepest descent means using negative gradient direction and exact line search on that direction.
But in this note, It seems as well as we are following negative gradient, the method can be called steepest descent..

What is the steepest descent method norm?

The direction of steepest descent is the unit vector that minimizes the directional derivative, i.e. d should minimize ⟨∇f,d⟩=∇fTAd.
This direction is when Ad=−∇f.
As A is positive definite we can solve this system to get d=u221.

1. A−1∇f

Where did steepest descent method come from?

Mathematicians have often attributed the method of steepest descent to the physicist Peter Debye, who in 1909 worked it out in an asymptotic study of Bessel functions.
Debye himself remarked that he had borrowed the idea of the method from an 1863 paper of Bernhard Riemann..

Why do we use steepest descent method?

The steepest descent method is one of the oldest known methods for minimizing a general nonlinear function.
The convergence theory for the method is widely used and is the basis for understanding many of the more sophisticated and well known algorithms..

1. A−1∇f